An expression for the quantum capacity of the widely used Pauli channels remains unknown [1]}, [2]}. However, a lower bound that can be achieved by stabilizer codes, the hashing bound, \(C_\mathrm {H}\) , [2]} is known. The reason why the quantum capacity of a Pauli channel can be higher than the hashing bound, i.e. \(C_\mathrm {Q} \ge C_\mathrm {H}\) , is the degenerate nature of quantum codes [4]}, [5]}, which arises from the fact that several distinct channel errors affect quantum states in an indistinguishable manner.
